A classification of dynamic programming formulations for offline deterministic single-machine scheduling problems

We review dynamic programming (DP) algorithms utilized to solve offline deterministic single-machine scheduling problems. We classify DP algorithms based on problem properties and provide insights on how these properties facilitate the use of specific types of DP algorithms. These properties center on whether jobs in a schedule can be naturally partitioned into subsets or whether a complete schedule is the outcome of blending distinct subsequences and/or whether the overall scheduling objective is a compromise of conflicting objectives. We propose generalizations of existing DP algorithms so they can be applied to more general problems such as proportionate flow shops. In some cases, we show how the running time of a DP algorithm can be improved. We also survey models where a DP formulation is part of a hybrid enumerative algorithm. A discussion on how pseudo-polynomial DP algorithms can be converted to fully polynomial time approximation schemes is also presented. We conclude our review with a timeline of DP algorithmic development during the last 50 years.